Stable node fixed point

Fig. 12.2.2. A nontransverse tangency of the unstable and strong-stable manifolds of a saddle-node fixed point may be obtained by a small time-periodic perturbation of the system with an on-edge homoclinic loop to a saddle-node equilibrium state, as shown in Fig. 12.1.4.

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

Fixed points boiling points of pure components and azeotropes. They can be nodes (stable and unstable) and saddles. 

A distillation boundary connects two fixed points node, stable or unstable, to a saddle. The distillation boundaries divide the separation space into separation regions. The shape of the distillation boundary plays an important role in the assessment of separations. 

The bifurcation at r is a saddle-node bifurcation, in which stable and unstable fixed points are bom out the clear blue sky as r is increased (see Section 3.1). 

If we continue to increase p, the stable and unstable fixed points eventually coalesce in a saddle-node bifurcation at // = 1. For p > 1 both fixed points have disappeared and now phase-locking is lost the phase difference 0 increases indefinitely, corresponding to phase drift (Figure 4.5.1c). (Of course, once 0 reaches 2jt the oscillators are in phase again.) Notice that the phases don t separate at a uniform rate, in qualitative agreement with the experiments of Hanson (1978) 0 increases most slowly when it passes under the minimum of the sine wave in Figure 4.5.1 c, at 0 = r/2, and most rapidly when it passes under the maximum at 0 = -kI2. ... 

A < 0. The parabola - 4A = 0 is the borderline between nodes and spirals star nodes and degenerate nodes live on this parabola. The stability of the nodes and spirals is determined by t. When t < 0, both eigenvalues have negative real parts, so the fixed point is stable. Unstable spirals and nodes have t > 0. Neutrally stable centers live on the borderline t = 0, where the eigenvalues are purely imaginary. 

Now because stable nodes and saddle points are not borderline cases, we can be certain that the fixed points for the full nonlinear system have been predicted correctly. 

Our example also illustrates some general mathematical concepts. Given an attracting fixed point x, we define its basin of attraction to be the set of initial conditions Xfl such that x(r) —> x as t —> w. For instance, the basin of attraction for the node at (3,0) consists of all the points lying below the stable manifold of the saddle. This basin is shown as the shaded region in Figure 6.4.8. 

Solution As shown previously, the system has four fixed points (0,0) = unstable node (0,2) and (3,0) = stable nodes and (1,1) = saddle point. The index at each of these points is shown in Figure 6.8.9. Now suppose that the system had a Q closed trajectory. Where could it lie ... 

Show that each of the following fixed points has an index equal to -bl. a) stable spiral b) unstable spiral c) center d) star e) degenerate node... 

Consider the phase portrait as p varies. For p>0. Figure 8.1.1 shows that there are two fixed points, a stable node at (x, y ) = (-, 0) and a saddle at (--Jp, 0). As p decreases, the saddle and node approach each other, then collide when p = 0, and finally disappear when p < 0. 

So A < 0 for the middle fixed point, which has 0 < x < 1 this is a saddle point. The fixed point with x > 1 is always a stable node, since A < afe and therefore... 

These equations can display a whole range of quantitatively different types of dynamics, including fixed points (nodes or stable foci), limit cycles, chaos, and quasi-periodicity. In this section we briefly describe these different types of... 

Pig. 1.7. Normalized fluctuations of the inter-spike interval versus the noise intensity for the FitzHugh-Nagmno model. Black curve reproduces the result shown in 40). Parameters 6o = 1.05, e = 0.01, 7 = 1,4, o = 1/3 in Eqs. 1.31. The fixed point is a stable focus. Same results with iip = 1.2 shown by the red curve, where the fixed point is a stable node. 

We choose the control parameters U and a such that the deterministic system exhibits no oscillations but is very close to a bifurcation thus yielding it very sensitive to noise. The transition from stationarity to oscillations in the system may occur either via a Hopf or via a saddle-node bifurcation on a limit cycle as depicted in the bifurcation diagram of Fig. 5.9. The different nature of these two bifurcations is reflected in the effect noise has in each case. The local character of the Hopf bifurcation is responsible for noise-induced high frequency oscillations of strongly varying amplitude around the stable fixed point. We try to characterize basic features of these oscillations such as coherence and time scales. The need to be able to adjust these features as one wishes will lead to the application of the time-delayed... 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

The theoretically predicted destruction of the resonance attractor in response to deviations from the circular shape of the integration domain has been confirmed experimentally within the light-sensitive BZ medium. A spiral wave was exposed to uniform illumination proportional to the total gray level obtained in an elliptical integration domain. Fig. 9.13(b) shows the resonant drift mediated during global feedback control. The spiral wave drifts towards a stable node of the drift velocity field. Close to this fixed point the drift velocity becomes very slow. Thus, the experimentally observed termination of the spiral drift at certain positions in a uniform medium is explained in the framework of the developed theory of feedback-mediated resonant drift. 

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

The fixed points obtained by solving (6.15) for a large number of iterates are stable nodes in the rectifying cascade. The same fixed points will form a subset of solu-... 

Equation (6.21) has the same fixed points as (6.20) except that their stability is reversed. Thus, a fixed point which is a stable node in equation (6.20) becomes an unstable node for equation (6.21). 

It is easily verified fhat the fixed point (1/2, 1/2) is a saddle point, i.e., is real and positive and is real and negative, for all values of m- The fixed point (0,0) undergoes a Hopf bifurcation at mh = 1/2. Condition (5.68) ensures that (0,0) is a stable fixed point. It is a node, if the discriminant is positive, see Sect. 1.2.2, which implies that... 

As an example, consider the residue curve map for a ternary system with a minimumboiling binary azeotrope of heavy (H) and light (L) species, as shown in Figure 7.23. There are four fixed points one unstable node at the binary azeotrope (A), one stable node at the vertex for the heavy species (H), and two saddles at the vertices of the light (L) and intermediate (I) species. 

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